Results: Plots and Tables

This report contains different plots and tables that may be relevant for analysing the results. Observe:

• The hypercube \(Y_N\) is placed in, is approx. \([0, 10000m]^p\), since for each subproblem \(Y_N^s \subset [0, 10000]^p\). That is, more subproblems create a larger hypercube for the master problem and the volume (number of integer points) grows exponentially in \(p\).
• Given \(|Y_N|\), we can find the hypercube as \[[l_1, u_1] \times \cdots \times [l_p, u_p],\] where \(l_i = \min_{y \in Y_N}(y_i)\) and \(u_i = \max_{y \in Y_N}(y_i)\).
• The number of integer points in the hypercube is \[\prod_{i = 1}^p (u_i - l_i + 1)\]

Statistics for alg1

Given a problem consisting of \(m\) subproblems with \(Y_N^s\) given for each subproblem \(s\), we use a filtering algorithm to find \(Y_N\) (alg1).

The following instance/problem groups are generated given:

• \(p=2,\ldots, 5\). [4 options]
• \(m=2,\ldots 5\) where \(m\) is the number of subproblems. [4 options]
• All subproblems have the same method config or half have method u and l. [4 options]
• Five instances for each config. [5 options]

Status

• 1174/1312 problems have been solved, i.e. 138 remaining.
• 1100/1174 problems have 5 instances solved for each configuration.
• 61/1174 have not been classified at all.
• 414/1113 have not been fully classified (only classified extreme).

Problems solved for the analysis

Note that the width of objective \(i = 1, \ldots p\), \(w_i = [l_i, u_i]\) should be approx. \(10000m\). Check:

## # A tibble: 4 × 6
##       m mean_width1 mean_width2 mean_width3 mean_width4 mean_width5
##   <dbl>       <dbl>       <dbl>       <dbl>       <dbl>       <dbl>
## 1     2      19245.      19221.      19213.      18996.      18690.
## 2     3      28760.      28800.      28689.      28479.      27847.
## 3     4      38300.      38353.      38153.      37744.      36642.
## 4     5      47725.      47977.      47979.      47112.      44478

Size of \(Y_N\)

What is \(|Y_N|\) given the different methods of generating the set of nondominated points for the subproblems?

## # A tibble: 4 × 3
##   method mean_card     n
##   <chr>      <dbl> <int>
## 1 l        200085.   230
## 2 m        168316.   285
## 3 u         80828.   300
## 4 ul        92155.   285

Does \(p\) have an effect?

## # A tibble: 16 × 4
## # Groups:   method [4]
##    method     p mean_card     n
##    <chr>  <dbl>     <dbl> <int>
##  1 l          2     5829.    60
##  2 m          2     6828.    80
##  3 u          2     1164.    80
##  4 ul         2     2920.    80
##  5 l          3    61381.    60
##  6 m          3   180435.    80
##  7 u          3    12475.    80
##  8 ul         3    26863.    80
##  9 l          4   370968.    60
## 10 m          4   301251.    70
## 11 u          4    79341.    75
## 12 ul         4   185223.    75
## 13 l          5   394576.    50
## 14 m          5   216391.    55
## 15 u          5   264717.    65
## 16 ul         5   199795.    50

Does \(m\) have an effect?

## # A tibble: 15 × 4
## # Groups:   method [4]
##    method     m mean_card     n
##    <chr>  <dbl>     <dbl> <int>
##  1 l          2     8173.    80
##  2 m          2     5688.    80
##  3 u          2     4201.    80
##  4 ul         2     4923.    80
##  5 l          3   166384.    80
##  6 m          3    90077.    80
##  7 u          3    37283.    80
##  8 ul         3    90425.    80
##  9 l          4   457927.    70
## 10 m          4   313521.    70
## 11 u          4   190675.    80
## 12 ul         4   151509.    70
## 13 m          5   333862.    55
## 14 u          5    94592.    60
## 15 ul         5   146013.    55

Let us try to fit the results using function \(y=c_1 s^{(c_2p)} m^{c_3p}\) (different functions was tried and this gave the highest \(R^2\)) for each method.

## # A tibble: 4 × 15
##   method fit    tidied   r.squared adj.r.squared sigma statistic   p.value    df
##   <chr>  <list> <list>       <dbl>         <dbl> <dbl>     <dbl>     <dbl> <dbl>
## 1 l      <lm>   <tibble>     0.781         0.779 1.04       406. 1.15e- 75     2
## 2 m      <lm>   <tibble>     0.701         0.699 1.22       331. 9.87e- 75     2
## 3 ul     <lm>   <tibble>     0.882         0.881 0.747     1056. 1.01e-131     2
## 4 u      <lm>   <tibble>     0.943         0.943 0.522     2452. 2.32e-185     2
## # ℹ 6 more variables: logLik <dbl>, AIC <dbl>, BIC <dbl>, deviance <dbl>,
## #   df.residual <int>, nobs <int>

## # A tibble: 4 × 4
##   method    c1     c2    c3
##   <chr>  <dbl>  <dbl> <dbl>
## 1 l      113.  0.0794 1.12 
## 2 m      131.  0.0734 1.04 
## 3 ul      35.4 0.113  1.09 
## 4 u       24.9 0.135  0.939

Relative size of \(Y_N\)

Nondominated points classification

We classify the nondominated points into, extreme, supported non-extreme and unsupported.

## # A tibble: 1 × 3
##   minPctEx avePctExt maxPctEx
##      <dbl>     <dbl>    <dbl>
## 1 0.000461    0.0465    0.330

## # A tibble: 4 × 4
##   method minPctEx avePctExt maxPctEx
##   <chr>     <dbl>     <dbl>    <dbl>
## 1 l      0.00966     0.0893    0.302
## 2 ul     0.00639     0.0721    0.330
## 3 m      0.000461    0.0209    0.147
## 4 u      0.00196     0.0133    0.104

Plots used in the paper